The invention described herein may be manufactured and used by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor.
The present invention relates to methods and apparatuses for correlating or establishing a statistical correlation of plural random variables, more particularly to methods and apparatuses for doing same in relation to physical phenomena such as impulsive signals or noise.
The alpha-stable distribution is an important area for investigation. One reason for the importance of the alpha-stable distribution is that it models impulsive noise. Another reason is that this statistical distribution is expected from superposition in natural processes. The parameter alpha, xcex1, is the characteristic exponent that varies over 0 less than xcex1xe2x89xa62. The alpha-stable distribution includes the Gaussian when xcex1=2. For xcex1 less than two, the distribution becomes more impulsive, more non-Gaussian in nature, and the tails of the distribution become thicker. This makes the alpha-stable distribution an attractive choice for modeling signals and noise having an impulsive nature. Also, from the generalized central limit theorem, the stable distribution is the only limiting distribution for sums of independent and identically distributed (IID) random variables (stability propert). If the individual distributions have finite variance, then the limiting distribution is Gaussian. For a less than two, the individual distributions have infinite variance. For detailed information, see the following two books, each of which is hereby incorporated herein by reference: C. L. Nikias and M. Shao, Signal Processing with Alpha-Stable Distributions and Applications, John Wiley and Sons, New York, N.Y., 1995; G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman and Hall, New York, N.Y., 1994.
Sources that could follow or be modeled by the alpha-stable distribution are abundant and include lightning in the atmosphere, switching transients in power lines, static in telephone lines, seismic activity, climatology and weather, ocean wave variability, surface texture, the slamming of a ship hull in a seaway, acoustic emissions from cracks growing in engineering materials under stress, magnetic avalanche or Barkhausen noise, transition boundary layer flow, etc. Many sources can exist in the area of target and background signatures that affect detection and classification. In underwater acoustics, examples of these sources could include interference to target detection such as ice cracking, biologics, bottom and sea clutter in active sonar, ocean waves near the surface and in the surf zone. They could also include target characteristics such as target strength in active sonar and cavitation. Similar sources in radar and infrared can include: ocean waves in the form of sea clutter and radar cross section (RCS); see R. D. Pierce, xe2x80x9cRCS Characterization using the alpha-stable distribution,xe2x80x9d Proc. 1996 IEEE National Radar Conference, Ann Arbor, Mich., May, 13-16, 1996, pp 154-159, hereby incorporated herein by reference. A 10 second long example of spiky, horizontally polarized (H-pol), radar sea clutter is shown herein in FIG. 1.
Two known and limitedly successful methodologies of establishing a statistical correlation, based on the alpha-stable distribution, are classical second-order correlation and covariation correlation. Neither second-order correlators nor covariation correlators have proven capable of obtaining consistent estimates of the relationship between two channels of noise characterized by impulsiveness. Furthermore, second-order correlators operate in a realm wherein alpha is equal to two. Covariation correlators operate in a realm wherein alpha is greater than or equal to one and less than or equal to two. Hence, when alpha is less than one (as would generally be the case, for instance, when the noise is extremely spikey, even more than the noise illustrated in FIG. 1), neither second-order correlators nor covariation correlators work to provide consistent results.
Of interest and incorporated herein by reference are the following United States patents: Nishimori U.S. Pat. No. 5,982,810 issued Nov. 9, 1999; Honkisz U.S. Pat. No. 5,787,128 issued Jul. 28, 1998; Kazecki U.S. Pat. No. 5,365,549 issued Nov. 15, 1994; Baron U.S. Pat. No. 4,860,239 issued Aug. 22, 1989; Horner U.S. Pat. No. 4,826,285 issued May 2, 1989; Zeidler et al. U.S. Pat. No. 4,355,368 issued Oct. 19, 1982Kaelin al. U.S. Pat. No. 4,234,883 issued Nov. 18, 1980; Baario U.S. Pat. No. 4,117,480 issued Sep. 26, 1978; Fletcher et al. U.S. Pat. No. 4,112,497 issued Sep. 05, 1978; Heng et al. U.S. Pat. No. 4,070,652 issued Jan. 24, 1978.
In view of the foregoing, it is an object of the present invention to provide an alpha-stable distribution based correlator which can reliably estimate the relationship between two channels of impulsive noise.
It is a further object of the present invention to provide an alpha-stable distribution-based correlator which can operate in a realm wherein alpha is less than one.
The present invention provides a methodology for correlating at least two remotely and/or locally generated signals, e.g., quantifying a relationship therebetween. An inventive method is provided for producing an output signal which is indicative of a correlation of at least two input signals. The inventive method comprises calculating (i.e., estimating) the codifference correlation with respect to at least two input signals, and producing an output signal commensurate with the codifference correlation. The estimating includes considering the sum and difference of codifference estimates wherein each codifference estimate is equated with a corresponding dispersion estimate. According to typical inventive practice, the calculating includes treating each signal as a complex signal representative of real and imaginary terms characterized by values and defined by real and imaginary axes.
Further provided according to this invention is a correlator for correlating a reference signal and a sampler signal. The inventive correlator admits of implementation in a communication system of the type including antenna means for receiving electromagnetic waves, down converter means for down converting a modulated signal received from the antenna, means, sampler means for sampling a down converted. signal received from the down converted means, and memory means for storing a reference signal. The inventive correlator correlates the reference signal (received from the memory means) and the sampler signal (received from the sampling means). The inventive correlator comprises algorithmic means for calculating (i.e., estimating) the codifference correlation based on the sum and difference of codifference estimates, wherein each codifference estimate is equated with a corresponding dispersion estimate.
This invention further provides a correlation detection system for use in association with a modulated signal such as produced by an antenna receiving electromagnetic waves such as radio frequency waves. The inventive correlation detection system comprises a down converter, a sampler, a memory and a correlator. The down converter is adaptable to down converting the modulated signal and producing a down converted signal. The sampler is adaptable to sampling the down converted signal and producing a sampled signal. The memory is adaptable to storing a reference signal. The correlator is adaptable to correlating the reference signal and the sampled signal. The correlator includes processor means and is capable of calculating (i.e., estimating) the, codifference correlation between the reference signal and the sampled signal, based on the sum and difference of codifference estimates. Each codifference estimate is taken from equation with a corresponding dispersion estimate.
In accordance with the present invention, the measure of a normalized codifference correlation can be indicated as h. An overall estimate h of the normalized codifference correlation between two signals xxe2x80x2 and yxe2x80x2 is found by the following equation, wherein h is equated to a quotient expression which takes the sum and difference of various codifference estimates:
ĥ=[({circumflex over (xcfx84)}xxe2x80x2R,xe2x88x92yxe2x80x2Rxe2x88x92{circumflex over (xcfx84)}xxe2x80x2R,yxe2x80x2R+{circumflex over (xcfx84)}xxe2x80x2I,xe2x88x92yxe2x80x2Ixe2x88x92{circumflex over (xcfx84)}xxe2x80x2I,yxe2x80x2I)+
i({circumflex over (xcfx84)}xxe2x80x2R,xe2x88x92yxe2x80x2Ixe2x88x92{circumflex over (xcfx84)}xxe2x80x2R,yxe2x80x2lxe2x88x92({circumflex over (xcfx84)}xxe2x80x2I,xe2x88x92yxe2x80x2Rxe2x88x92{circumflex over (xcfx84)}xxe2x80x2I,yxe2x80x2R))]/(2xcex1{circumflex over (xcex3)}xxe2x80x2)
The codifference estimates are each formed from its corresponding dispersion estimate, as follows:
{circumflex over (xcfx84)}xxe2x80x2R,xe2x88x92yxe2x80x2R={circumflex over (xcex3)}xxe2x80x2R+{circumflex over (xcex3)}yxe2x80x2Rxe2x88x92{circumflex over (xcex3)}xxe2x80x2R+yxe2x80x2R
{circumflex over (xcfx84)}xxe2x80x2R,yxe2x80x2R={circumflex over (xcex3)}xxe2x80x2R+{circumflex over (xcex3)}yxe2x80x2Rxe2x88x92{circumflex over (xcex3)}xxe2x80x2Rxe2x88x92yxe2x80x2R
{circumflex over (xcfx84)}xxe2x80x2I,xe2x88x92yxe2x80x2I={circumflex over (xcex3)}xxe2x80x2I+{circumflex over (xcex3)}xxe2x80x2I+yxe2x80x2I
{circumflex over (xcfx84)}xxe2x80x2I,yxe2x80x2I={circumflex over (xcex3)}xxe2x80x2I+{circumflex over (xcex3)}yxe2x80x2Ixe2x88x92{circumflex over (xcex3)}xxe2x80x2Ixe2x88x92yxe2x80x2I
{circumflex over (xcfx84)}xxe2x80x2R,xe2x88x92yxe2x80x2I={circumflex over (xcex3)}xxe2x80x2R+{circumflex over (xcex3)}yxe2x80x2Ixe2x88x92{circumflex over (xcex3)}xxe2x80x2R+yxe2x80x2I
{circumflex over (xcfx84)}xxe2x80x2R,yxe2x80x2I={circumflex over (xcex3)}xxe2x80x2R+{circumflex over (xcex3)}yxe2x80x2Ixe2x88x92{circumflex over (xcex3)}xxe2x80x2Rxe2x88x92yxe2x80x2I
{circumflex over (xcfx84)}xxe2x80x2I,xe2x88x92yxe2x80x2R={circumflex over (xcex3)}xxe2x80x2I+{circumflex over (xcex3)}yxe2x80x2Rxe2x88x92{circumflex over (xcex3)}xxe2x80x2I+yxe2x80x2R
{circumflex over (xcfx84)}xxe2x80x2I,yxe2x80x2R={circumflex over (xcex3)}xxe2x80x2I+{circumflex over (xcex3)}yxe2x80x2Rxe2x88x92{circumflex over (xcex3)}xxe2x80x2Ixe2x88x92yxe2x80x2R
using the real and imaginary terms from the input (or first signal)
xxe2x80x2=xxe2x80x2R+ixxe2x80x2I=(xR+nR)+i(xI+nI)
and the real and imaginary terms from the output (or second signal)
yxe2x80x2=yxe2x80x2R+iyxe2x80x2I=(hRxRxe2x88x92hIxI+mR)+i(hIxR+hRxI+mI)
Using xxe2x80x2R as an example, each of the dispersions is estimated from the N data samples             γ      ^              x      R      xe2x80x2        =                    (                  C          ⁢                      (                          p              ,              α                        )                          )                              -          α                /        p              ⁢                  (                              1            N                    ⁢                                    ∑                              k                =                1                            N                        ⁢                          xe2x80x83                        ⁢                                          "LeftBracketingBar"                                                      x                    R                    xe2x80x2                                    ⁢                                      (                    k                    )                                                  "RightBracketingBar"                            p                                      )                    α        /        p            
where       C    ⁢          (              p        ,        α            )        =            2      p        ⁢                            Γ          ⁢                      (                                          p                +                1                            2                        )                          ⁢                  Γ          ⁢                      (                          1              -                              p                α                                      )                                                Γ          ⁢                      (                          1              2                        )                          ⁢                  Γ          ⁢                      (                          1              -                              p                2                                      )                              
and
xe2x88x921 less than p less than xcex1
where p can be selected from this interval using a minimum error procedure for a given alpha, and where the normalizing term in the codifference correlator is given by
{circumflex over (xcex3)}xxe2x80x2={circumflex over (xcex3)}xxe2x80x2R+{circumflex over (xcex3)}xxe2x80x2I.
Incorporated herein by reference is McLaughlin,. D. J., et al, xe2x80x9cHigh Resolution Polarimetric Radar Scattering Measurements of Low Grazing Angle Sea Clutter,xe2x80x9d IEEE Journal of Oceanic Engineering, Vol. 20, No. 3, July 1995, pp 166-178. An illustrative application of the inventive unnormalized codifference correlator is represented by the calculation of the terms for the averaged Mueller matrix as shown in McLaughlin equation (8). The Mueller matrix is used to describe the polarimetric scattering behavior of a radar target. Here the four complex-valued terms from the polarization scattering matrix (McLaughlin equation (1b)) are taken a pair at a time, and temporally (time) or spatially averaged to cover all sixteen combinations. This operation utilizes the classical second-order correlator (unnormalized). When applied to radar, scattering from spiky sea clutter, the unnormalized codifference correlator in accordance with the present invention would be expected to give a more accurate and consistent measure of the terms for the Mueller matrix.
The unnormalized codifference correlator is given by
{circumflex over (q)}xxe2x80x2yxe2x80x2=({circumflex over (xcfx84)}xxe2x80x2R,xe2x88x92yxe2x80x2Rxe2x88x92{circumflex over (xcfx84)}xxe2x80x2R,yxe2x80x2R+{circumflex over (xcfx84)}xxe2x80x2I,xe2x88x92yxe2x80x2Ixe2x88x92{circumflex over (xcfx84)}xxe2x80x2I,yxe2x80x2I)+
i({circumflex over (xcfx84)}xxe2x80x2R,xe2x88x92yxe2x80x2Ixe2x88x92{circumflex over (xcfx84)}xxe2x80x2R,yxe2x80x2Ixe2x88x92({circumflex over (xcfx84)}xxe2x80x2I,xe2x88x92yxe2x80x2Rxe2x88x92{circumflex over (xcfx84)}xxe2x80x2I,yxe2x80x2R))
where
xxe2x80x2=xxe2x80x2R+ixxe2x80x2I
yxe2x80x2=yxe2x80x2R+iyxe2x80x2I
And, when related to McLaughlin equation (8) by using, for example, the average  less than S*vhSvv greater than , the averaged product is replaced by the unnormalized codifference correlator using the polarization scattering elements where
xxe2x80x2=Svv
yxe2x80x2=Svh
For symmetric alpha-stable (Sxcex1S) random variables with any characteristic exponent, alpha, from zero to two, the codifference is a measure of bivariate dependence. Signals and noise with this statistical distribution are impulsive in nature and classical correlation methods can give inconsistent answers. In accordance with the present invention, the properties of the codifference are exploited to construct a codifference correlator that can be used for estimating or quantifying the relationship between two complex, isotropic Sxcex1S random variables.
The traditional covariation correlator only works for alpha from one to two. The traditional classical correlator uses covariance (correlation coefficients and transfer functions) which, for Sxcex1S signal and noise, is only defined for alpha of two. The present invention""s codifference correlator functions in a manner somewhat comparable to the covariation correlator and the classical correlator.
Besides including alpha from zero to two, a major advantage of the inventive codifference correlator is robustness to uncorrelated Sxcex1S noise added to both random variables. A disadvantage of the inventive codifference correlator is a nonlinear relationship or bias with respect to the xe2x80x9camount of dependence,xe2x80x9d especially for alpha less than one. Potential applications of the present invention can range from characterizing small radar targets in spiky sea, clutter (e.g., as illustrated in FIG. 1) to the possible estimation of boundary layer velocity and pressure relationships in transitional flow.
Under real world circumstances in which a common signal along with uncorrelated, additive noise having an impulsive character is present in each channel prior to measurement, the inventive codifference correlator is capable of obtaining a consistent estimate of the relationship or common signal between the two channels. Known methodologies (e.g., second-order correlators and covariation correlators) cannot provide consistent estimates of this relationship. The present invention""s codifference correlator will also work when the noise is extremely spiky (alpha less than one), and the known methodologies are not defined.